Abstract
A quantum computer will require quantum bits (qubits) with good coherence that can be coupled together to form logic gates^{1,2}. Superconducting circuits offer a novel solution^{3,4,5,6,7,8,9} because qubits can be connected in elaborate ways through simple wiring, much like that of conventional integrated circuits. However, this ease of coupling is offset by coherence times shorter than those observed in molecular and atomic systems. Hybrid architectures could help skirt this fundamental tradeoff between coupling and coherence by using macroscopic qubits for coupling and atombased qubits for coherent storage^{10,11}. Here, we demonstrate the first quantum memory operation^{12} on a Josephsonphase qubit by transferring an arbitrary quantum state to a twolevel state^{13} (TLS), storing it there for some time, and later retrieving it. The qubit is used to probe the coherence of the TLS by measuring its energy relaxation and dephasing times. Quantum process tomography^{2,14} completely characterizes the memory operation, yielding an overall process fidelity of 79%. Although the uncontrolled distribution of TLSs precludes their direct use in a scalable architecture, the ability to coherently couple a macroscopic device with an atomicsized system motivates a search for designer molecules that could replace the TLS in future hybrid qubits.
Main
In quantum computation, coupling atomic qubits over macroscopic distances is a longstanding technological challenge. In iontrap architectures, qubits are physically moved to regions where they can be positioned close to each other and coupled electrostatically^{15}. Approaches based on cavity QED eliminate the difficulty of moving atoms, but instead use a resonant electromagnetic cavity to couple over macroscopic distances via guided photons^{5,6,16}. Several recent proposals meet this challenge using further novel approaches^{10,11,17,18}.
Superconducting wires are a natural medium for coupling between macroscopic and atomic states because currents and voltages obey quantum mechanics over length scales from macroscopic to atomic dimensions. At the macroscopic scale, the coupling remains coherent because superconductors have small dissipation. At the atomic scale, coupling is possible because the tunnel junction has a dielectric thickness ∼2 nm that approaches atomic size. When an atom carrying a single elementary charge moves by one atomic bond length inside such a tunnel junction, it produces a substantial image charge in the junction electrodes, coupling the atomicscale motion to the macroscopic degrees of freedom of the currents and voltages in the circuit. We thus have a natural hybrid system: the atomic state in the junction is a ‘memory’ qubit capable of storing a quantum state, whereas the Josephson junction itself is a ‘register’ qubit capable of general logic operations and able to be coupled to other qubits over macroscopic distances.
Our register qubit, a fluxbiased Josephsonphase qubit, is shown schematically in Fig. 1a. The qubit is a nonlinear inductor–capacitor resonator, whose resonant frequency can be tuned by varying the magnetic flux applied to the loop. The nonlinear resonator is well modelled at typical flux biases with a cubic potential, as shown in Fig. 1b. The nonlinearity breaks the degeneracy (equal spacing) between adjacent energy levels, so that the application of microwaves produces transitions between only one pair of quantum states. Experiments are carried out so that only the two lowestlying states are occupied; these constitute the qubit states 0〉 and 1〉. The qubit is controlled by applying pulses of magnetic flux. A quasid.c. pulse adiabatically changes the energy difference between 0〉 and 1〉, and the resulting accumulation of phase is equivalent to a Zaxis rotation of the Bloch vector^{2}. A microwave pulse at the transition frequency coherently changes the occupation of the state, and corresponds to an X or Yaxis rotation^{19}.
Our memory qubit is a twolevel state (TLS) located inside the Josephson tunnel barrier, shown in Fig. 1c. A TLS is understood to be an atom, or a small group of atoms, that tunnels between two lattice configurations^{20}. Such states are ubiquitous in amorphous materials, the disordered structure of which does not fully constrain the constituent atoms. The electric dipole moment inferred from measurements of these states is consistent with an atom carrying net charge e moving by about one atomic bond length^{21}. Because the barrier thickness is ∼20 atoms, the induced charge on the electrodes from this motion is large, ∼e/20.
TLS defects have been considered deleterious, as they provide extra quantum states to which the qubit may couple in an uncontrolled manner. A dense bath of these states is equivalent to dielectric loss, which decreases the lifetime of the qubit^{21}. Decoherence from TLS defects can be removed by decreasing their number; this has been accomplished by reducing the junction area while shunting it with an external lowloss capacitor^{22}.
Here, we demonstrate that a TLS can play a useful role as well. A previous experiment coupled a qubit to a TLS^{13}, and theoretical work has suggested their use as memory elements^{12}. In the present device, individual TLS states are well separated from each other in frequency, and the qubit lifetime is sufficiently long, that we can carry out precise gate operations between the register qubit and one TLS. The ground and excited energy eigenstates of this TLS, labelled g〉 and e〉, constitute the memory qubit. By adjusting the flux bias, the register and memory qubits are tuned into and out of resonance, effectively turning on and off their coupling. We have found that a TLS memory can have a reasonably long coherence time, and thus represents a good model system for future hybrid qubits.
The register qubit transition frequency 0〉→1〉 is first measured as a function of flux bias using spectroscopy, as shown in Fig. 2. The splitting at ∼7.05 GHz is due to a TLS with coupling strength S=41 MHz that will be used as the memory qubit. We characterize the register qubit at the offresonance frequency of 6.75 GHz, and find coherence times from standard energy decay, Ramsey and spinecho sequences to be T_{1}=400 ns, T_{2}=120 ns and T_{2}^{*}=350 ns, respectively. The measurement visibility is high, approximately 90%. This phase qubit has a coherence time T_{1} that is four times longer than previously reported, owing to the use of a new lowloss dielectric aSi:H (refs 21, 23) in the shunt capacitor.
To characterize the memory qubit, we first tune the register qubit offresonance to 6.75 GHz and excite it into the 1〉 state with a 16nslong X_{π} pulse. Then, a Z pulse with 2 ns rise time moves the register qubit adiabatically into resonance with the memory qubit, effectively turning on the coupling. After waiting for time t, the register qubit state is measured. The resulting oscillations between the register and memory qubits, shown in Fig. 3a, have a 25 ns period that agrees with the coupling strength measured spectroscopically. In the rotating frame, the coupling is of the form^{12} (S/2) (σ_{x}σ_{x}+σ_{y}σ_{y}), so that the first minimum at 12 ns corresponds to an iSWAP gate^{24}, which takes 1g〉→i0e〉 and 0e〉→i1g〉. The envelope of the oscillations between the register and memory decays more slowly than for the register qubit alone, indicating that the memory qubit has a longer lifetime.
The coherence of the memory qubit is directly measured using two iSWAP operations, as shown in Fig. 3b,c. We start by exciting the register qubit offresonance, and then move it into resonance with the memory qubit for time t_{swap}=12 ns to achieve state transfer into memory. After the register qubit is moved out of resonance to its starting frequency for a variable wait time t, it is then subjected to another iSWAP operation before measurement. Figure 3b shows a plot of the measurement probability versus wait time; the exponential decay gives a qubit memory time T_{1}=1.2 μs. Storing the quantum state in memory instead of the register increases its lifetime by a factor of three in our system, although this does not improve on the best T_{1} times reported in other superconducting qubits. TLS lifetimes estimated from phonon radiation^{25} depend on the electron–phonon coupling constant, which varies greatly from defect to defect. The measured lifetime is consistent with typical predictions that fall in the range of 10 ns–10 μs.
A similar Ramsey fringe experiment is used to measure dephasing, as shown in Fig. 3c. We first prepare the register qubit in the superposition with an X_{π/2} pulse, carry out the same iSWAP/hold/iSWAP sequence as before, and then execute a final π/2 pulse with swept phase. The envelope of the Ramsey oscillations indicates a memory dephasing time T_{2}=210 ns.
This iSWAP/hold/iSWAP sequence is in fact a quantum memory operation for an arbitrary initial state in the register qubit. The first iSWAP transfers the state to the memory qubit, where it is protected from decoherence during the hold time; the second iSWAP then restores the register qubit to its initial state (up to a correctable Z rotation). We characterize this memory operation using quantum process tomography^{2,14} (QPT), which involves preparing a spanning set of input states, carrying out the quantum operation and measuring the output with quantum state tomography^{22} (QST). The measured input and output states enable us to fully reconstruct the quantum memory process. The control sequence for QPT is similar to that for Ramsey fringes, with the microwave pulses generalized to create the initial states and to carry out QST on the final states.
QST is carried out at three separate stages in the sequence, as shown in Fig. 4a–c. In Fig. 4b, after transfer to the memory qubit, the register qubit contains little amplitude of the initial state, as expected. The coherence of the memory operation is determined by comparing Fig. 4a–c, which shows only a small reduction in length of the Bloch vectors. Here, we have compensated for the Z rotation arising from the two iSWAPs and the 295 MHz detuning between the coupling ‘on’ and ‘off’ frequencies.
QPT gives us the χmatrix of the memory operation^{2}, shown in Fig. 4d. In this representation, the quantum operation acts on the input density matrix as , where is some fixed set of basis operators, in our case the identity (I) and Pauli σmatrices. Diagonalizing the χmatrix leads to the operatorsum representation, which we write as , where the operation elements are linear combinations of the basis operators, and the weights {w_{k}} give the probabilities of applying each operation element. Table 1 shows the operatorsum representation of the memory operation, giving the weight w_{k} and the coefficients of the basis operators for each operation element . The dominant operation element is a nearidentity, as we expect for a memory operation. The nextmost dominant elements are primarily σ_{z}like (T_{2} dephasing) and σ_{x} and σ_{y}like (T_{1} relaxation). The relative weights of these dephasing and relaxation elements are roughly as expected from the measured T_{1} and T_{2} times of the memory qubit, accounting for the overall length of the experiment ∼40 ns. The simplest measure of fidelity, the trace overlap, gives a process fidelity of 79%.
Errors in the memory operation can be divided into several categories, including tomography errors during state preparation and measurement, storage errors during the memory hold time and transfer errors during the iSWAPs. Tomography errors will be reduced by ongoing work to improve singlequbit performance through, for example, better materials and microwave pulse shaping. Storage errors represent the intrinsic T_{1} and T_{2} of the TLS memory qubit, and it may be possible to improve them by substituting a suitably engineered molecule. Finally, transfer errors that come from the register–memory interaction may be more difficult to reduce, although careful control of the qubit frequency in turning on and off the coupling should improve the transfer fidelity. Note that the transfer errors are independent of the memory hold time. Thus, after some crossover time t_{c}, the longer T_{1} of the memory qubit offsets the transfer error, resulting in better overall memory fidelity than the register qubit alone. In this case, our analysis of the process tomography indicates that the TLS memory is useful beyond the crossover time t_{c}∼ 100 ns.
Although a TLS was suitable for this initial proofofprinciple demonstration of quantum memory, their use in a quantum computer is unlikely because of their intrinsically random nature and limited coherence time. However, this experiment explicitly demonstrates a bridge between macroscopic and atomicsized qubit states in a Josephson qubit, which motivates a search for a properly engineered atomicscale memory qubit. Such a molecule should have a transition frequency in the 3–15 GHz microwave range, a large electric dipole moment to couple to the capacitor, but a small motional dipole moment to minimize coupling to and loss from phonon radiation. With the increasing ability in the field of molecular electronics to fabricate designer molecules, a hybrid Josephson qubit with long coherence may be within reach.
Methods
Fabrication
The phase qubit used in this experiment was fabricated using procedures similar to those described in ref. 22. We replaced the silicon nitride dielectric of that design with a new lowloss dielectric made from hydrogenated amorphous silicon to achieve ∼4 times longer coherence time.
Qubit control
Control signals for carrying out qubit manipulations are generated using a custom 2channel 14bit digitaltoanalog converter with 1 ns waveform resolution. The output of the digitaltoanalog converter is filtered with dissipative gaussian filters. Quasid.c. pulses for Z rotations and measurement are sent directly to the qubit, whereas the microwave control signals are sent to a quadrature mixer to modulate the two quadratures of a microwave signal from a continuouswave source. The input microwave frequency is set 100 MHz above the qubit frequency, and a sideband modulation is used to mix this signal into resonance with the qubit. This prevents leakage at the carrier frequency from causing qubit transitions, thereby increasing the on/off ratio of the microwave control.
State tomography
Three suitably chosen measurements are sufficient to completely characterize a singlequbit state, for example, projections along the X, Y and Z axes of the Bloch sphere. The phase qubit can only be measured along the Z axis (distinguishing 0〉and 1〉), but rotations can be applied before measurement to effectively measure along other axes. For the present device, the visibility of the 0〉 and 1〉 states is different, that is, the probability of correctly identifying 0〉 is not equal to the probability of correctly identifying 1〉. For this reason, we carried out six measurements, one in each direction along the X, Y and Z axes. These six measurements can be combined to yield the Bloch vector of the state.
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Acknowledgements
Devices were made at the UCSB and Cornell Nanofabrication Facilities, a part of the NSF funded NNIN network. This work was supported by ARDA under grant W911NF0410204 and NSF under grant CCF0507227.
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Neeley, M., Ansmann, M., Bialczak, R. et al. Process tomography of quantum memory in a Josephsonphase qubit coupled to a twolevel state. Nature Phys 4, 523–526 (2008). https://doi.org/10.1038/nphys972
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